Global uniqueness of the rotational shooting root

Determine whether, for fixed R > 0 and c > 0 satisfying cR^2 ≤ 2, the shooting equation Q(a) = 0 has a unique root a in the full shooting domain A; in particular, determine whether, in the perturbative regime of Theorem 1.3, the root continued from the critical-catenoid root is the only root in A.

Background

The paper formulates rotational Gaussian f-minimal annuli through the profile ODE (7.1), with neck radius a, first sphere-hitting time S_a, shooting domain A defined by finite transversal hitting, and shooting map Q(a) given by the free-boundary residual. A rotational free-boundary Gaussian f-minimal annulus corresponds precisely to a root of Q in A.

Theorem 1.3 establishes, for sufficiently small positive c and fixed R, a root locally continued from the nondegenerate critical-catenoid root at c = 0, and proves uniqueness only within the neighborhood where the implicit-function theorem applies. The unresolved issue is whether additional roots occur elsewhere in the full shooting domain A. A positive answer to global uniqueness would, via Proposition 7.5, yield uniqueness up to ambient rotations for embedded annuli in the non-disk branch.